Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2211.10993 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- Given a lattice polytope $Q\subset \mathbb{R}^n$, we can consider the cone $σ=C(Q)=\{λ(q,1)\in \mathbb{R}^{n+1}|λ\in \mathbb{R}_{\geq0}, q\in Q\} \subset \mathbb{R}^{n+1}$, and the affine toric variety $Y_σ$ associated to $σ$. Altmann showed that the versal deformation space of $Y_σ$ can be described by the Minkowski decomposition of the polytope $Q$. Under some conditions on $Q$, we can obtain a smooth deformation $Y_ε$ of $Y_σ$ using Altmann's result. In this article, we consider $Y_ε$ inside some $\mathbb{C}^N$ and construct a complex fibration on $Y_ε$, with general fibre $(\mathbb{C}^*)^n$ and finite singular fibres described using global coordinates related to the components of the Minkowski decomposition. We construct a singular Lagrangian torus fibration out of the complex fibration. This singular fibration admits a convex base diagram representation with cuts as a natural generalization of base diagrams described by Symington for Almost Toric Fibrations ($\dim=4$). In particular, we obtain a convex base diagram whose image is the dual cone of $C(Q)$. There is a 1-parameter family of monotone Lagrangian tori in each of these fibrations. Using the wall-crossing formula, we describe the potential associated with this family in terms of the Minkowski decomposition of $Q$, recovering the result of Lau, and discuss non-displaceability. We also discuss some other consequences of our results.